SHP（光滑粒子流体动力学）流体模拟实现三：Marching Cube算法（1）

流体模拟（三）

Marching Cube算法（1）

我们在 实现流体表面重建时，需要事先在空间中划分网格，我们的流体系统正好已经完成了此项工作。其次利用Marching Cube算法计算出构成表面的三角片面，最后根据所有三角片面的计算每个顶点的法线值。

Marching Cube算法，最早是Lorensen等人在“Marching cubes: A high resolution 3D surface construction algorithm”这篇文章中提出的算法，他们最早是用来实现医学图像的三维重建，现在该算法可以用于任何基于标量场的表面重建，并还有许多优化版本。本文就提一下它的最初版本。

我们给空间划分为了许多小网格，每个网格也称为体元（cell），如下图所示：

由于该算法是基于标量场的，所以我们需要知道网格点是在场外还是场内（表面外还是表面内），即f(x,y,z)=0的点为表面点，所以也称点(x,y,z)满足该隐函数。由于每个体元上有八个顶点。若连续的两个顶点，有一个顶点位于表面外，另一个顶点位于表面内，则我们的表面就肯定位于这两个顶点之间，即可以用加权插值获得最后表面的位置。因此我们的算法需要遍历所有的网格，判断网格的每个顶点是在表面外还是表面内，若为表面内则赋值1，否则赋值0。

Lorensen等人他们就发现了每个网格一共就8个顶点，那所有的可能性就只有=256种。并且由于立方体具有旋转对称性，经过一定的旋转会得到同样的值，因此他们经过归纳发现所有情况可以归结为以下15种：

 之后给每个立方体定义了一个固定的顶点编号和边编号：

如图所示，v为顶点编号e为边编号。因为正好是八个顶点，就可以利用一个char类型（8位）来保存该编号索引。由于我们最后的三角片面的顶点都会在边上，立方体有12条边，所以我们可以用一个12位2进制数表示边是否含有三角片面的点。因为我们构建的三角形片面情况一共只有256种，所以我们可以建立边索引表。如我们顶点索引位00000001，说明只有v1在表面内，因此说明表面的顶点在e1，e4，e9上，则边索引为001000001001，化为16进制为0x109。以顶点的八位二进制值为边表索引，以对应的12位二进制值为边表值，则上述例子可表示为边表索引为1的值为0x109。然后对256种情况一一对应即可构造出边表。构造的边表如下：

const unsigned int edgeTable[256] = {
    0x0  , 0x109, 0x203, 0x30a, 0x406, 0x50f, 0x605, 0x70c,
    0x80c, 0x905, 0xa0f, 0xb06, 0xc0a, 0xd03, 0xe09, 0xf00,
    0x190, 0x99 , 0x393, 0x29a, 0x596, 0x49f, 0x795, 0x69c,
    0x99c, 0x895, 0xb9f, 0xa96, 0xd9a, 0xc93, 0xf99, 0xe90,
    0x230, 0x339, 0x33 , 0x13a, 0x636, 0x73f, 0x435, 0x53c,
    0xa3c, 0xb35, 0x83f, 0x936, 0xe3a, 0xf33, 0xc39, 0xd30,
    0x3a0, 0x2a9, 0x1a3, 0xaa , 0x7a6, 0x6af, 0x5a5, 0x4ac,
    0xbac, 0xaa5, 0x9af, 0x8a6, 0xfaa, 0xea3, 0xda9, 0xca0,
    0x460, 0x569, 0x663, 0x76a, 0x66 , 0x16f, 0x265, 0x36c,
    0xc6c, 0xd65, 0xe6f, 0xf66, 0x86a, 0x963, 0xa69, 0xb60,
    0x5f0, 0x4f9, 0x7f3, 0x6fa, 0x1f6, 0xff , 0x3f5, 0x2fc,
    0xdfc, 0xcf5, 0xfff, 0xef6, 0x9fa, 0x8f3, 0xbf9, 0xaf0,
    0x650, 0x759, 0x453, 0x55a, 0x256, 0x35f, 0x55 , 0x15c,
    0xe5c, 0xf55, 0xc5f, 0xd56, 0xa5a, 0xb53, 0x859, 0x950,
    0x7c0, 0x6c9, 0x5c3, 0x4ca, 0x3c6, 0x2cf, 0x1c5, 0xcc ,
    0xfcc, 0xec5, 0xdcf, 0xcc6, 0xbca, 0xac3, 0x9c9, 0x8c0,
    0x8c0, 0x9c9, 0xac3, 0xbca, 0xcc6, 0xdcf, 0xec5, 0xfcc,
    0xcc , 0x1c5, 0x2cf, 0x3c6, 0x4ca, 0x5c3, 0x6c9, 0x7c0,
    0x950, 0x859, 0xb53, 0xa5a, 0xd56, 0xc5f, 0xf55, 0xe5c,
    0x15c, 0x55 , 0x35f, 0x256, 0x55a, 0x453, 0x759, 0x650,
    0xaf0, 0xbf9, 0x8f3, 0x9fa, 0xef6, 0xfff, 0xcf5, 0xdfc,
    0x2fc, 0x3f5, 0xff , 0x1f6, 0x6fa, 0x7f3, 0x4f9, 0x5f0,
    0xb60, 0xa69, 0x963, 0x86a, 0xf66, 0xe6f, 0xd65, 0xc6c,
    0x36c, 0x265, 0x16f, 0x66 , 0x76a, 0x663, 0x569, 0x460,
    0xca0, 0xda9, 0xea3, 0xfaa, 0x8a6, 0x9af, 0xaa5, 0xbac,
    0x4ac, 0x5a5, 0x6af, 0x7a6, 0xaa , 0x1a3, 0x2a9, 0x3a0,
    0xd30, 0xc39, 0xf33, 0xe3a, 0x936, 0x83f, 0xb35, 0xa3c,
    0x53c, 0x435, 0x73f, 0x636, 0x13a, 0x33 , 0x339, 0x230,
    0xe90, 0xf99, 0xc93, 0xd9a, 0xa96, 0xb9f, 0x895, 0x99c,
    0x69c, 0x795, 0x49f, 0x596, 0x29a, 0x393, 0x99 , 0x190,
    0xf00, 0xe09, 0xd03, 0xc0a, 0xb06, 0xa0f, 0x905, 0x80c,
    0x70c, 0x605, 0x50f, 0x406, 0x30a, 0x203, 0x109, 0x0
};

构造完边表之后，我们还需要构造一个三角形表。三角形表记录了体元中三角形片面的连接方式。根据那15种情况，可以发现一个体元内的三角面最多只有4个，所以我们的三角形表可以定义为一个256*12的2维数组。同样通过顶点索引出三角形连接方式。如同样的0000001。则triTable[1]={0, 8, 3, X, X, X, X, X, X, X, X, X, X, X, X, X}（X定义为255即不存在点）。表示当三角形在顶点v1在表面内时，只有一个三角形，且三角形的三个点在边e1，e9，e4上。我们贴上三角形表的代码：

#define X 255
const int triTable[256][16] = {
    {X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X},
    {0, 8, 3, X, X, X, X, X, X, X, X, X, X, X, X, X},
    {0, 1, 9, X, X, X, X, X, X, X, X, X, X, X, X, X},
    {1, 8, 3, 9, 8, 1, X, X, X, X, X, X, X, X, X, X},
    {1, 2, 10, X, X, X, X, X, X, X, X, X, X, X, X, X},
    {0, 8, 3, 1, 2, 10, X, X, X, X, X, X, X, X, X, X},
    {9, 2, 10, 0, 2, 9, X, X, X, X, X, X, X, X, X, X},
    {2, 8, 3, 2, 10, 8, 10, 9, 8, X, X, X, X, X, X, X},
    {3, 11, 2, X, X, X, X, X, X, X, X, X, X, X, X, X},
    {0, 11, 2, 8, 11, 0, X, X, X, X, X, X, X, X, X, X},
    {1, 9, 0, 2, 3, 11, X, X, X, X, X, X, X, X, X, X},
    {1, 11, 2, 1, 9, 11, 9, 8, 11, X, X, X, X, X, X, X},
    {3, 10, 1, 11, 10, 3, X, X, X, X, X, X, X, X, X, X},
    {0, 10, 1, 0, 8, 10, 8, 11, 10, X, X, X, X, X, X, X},
    {3, 9, 0, 3, 11, 9, 11, 10, 9, X, X, X, X, X, X, X},
    {9, 8, 10, 10, 8, 11, X, X, X, X, X, X, X, X, X, X},
    {4, 7, 8, X, X, X, X, X, X, X, X, X, X, X, X, X},
    {4, 3, 0, 7, 3, 4, X, X, X, X, X, X, X, X, X, X},
    {0, 1, 9, 8, 4, 7, X, X, X, X, X, X, X, X, X, X},
    {4, 1, 9, 4, 7, 1, 7, 3, 1, X, X, X, X, X, X, X},
    {1, 2, 10, 8, 4, 7, X, X, X, X, X, X, X, X, X, X},
    {3, 4, 7, 3, 0, 4, 1, 2, 10, X, X, X, X, X, X, X},
    {9, 2, 10, 9, 0, 2, 8, 4, 7, X, X, X, X, X, X, X},
    {2, 10, 9, 2, 9, 7, 2, 7, 3, 7, 9, 4, X, X, X, X},
    {8, 4, 7, 3, 11, 2, X, X, X, X, X, X, X, X, X, X},
    {11, 4, 7, 11, 2, 4, 2, 0, 4, X, X, X, X, X, X, X},
    {9, 0, 1, 8, 4, 7, 2, 3, 11, X, X, X, X, X, X, X},
    {4, 7, 11, 9, 4, 11, 9, 11, 2, 9, 2, 1, X, X, X, X},
    {3, 10, 1, 3, 11, 10, 7, 8, 4, X, X, X, X, X, X, X},
    {1, 11, 10, 1, 4, 11, 1, 0, 4, 7, 11, 4, X, X, X, X},
    {4, 7, 8, 9, 0, 11, 9, 11, 10, 11, 0, 3, X, X, X, X},
    {4, 7, 11, 4, 11, 9, 9, 11, 10, X, X, X, X, X, X, X},
    {9, 5, 4, X, X, X, X, X, X, X, X, X, X, X, X, X},
    {9, 5, 4, 0, 8, 3, X, X, X, X, X, X, X, X, X, X},
    {0, 5, 4, 1, 5, 0, X, X, X, X, X, X, X, X, X, X},
    {8, 5, 4, 8, 3, 5, 3, 1, 5, X, X, X, X, X, X, X},
    {1, 2, 10, 9, 5, 4, X, X, X, X, X, X, X, X, X, X},
    {3, 0, 8, 1, 2, 10, 4, 9, 5, X, X, X, X, X, X, X},
    {5, 2, 10, 5, 4, 2, 4, 0, 2, X, X, X, X, X, X, X},
    {2, 10, 5, 3, 2, 5, 3, 5, 4, 3, 4, 8, X, X, X, X},
    {9, 5, 4, 2, 3, 11, X, X, X, X, X, X, X, X, X, X},
    {0, 11, 2, 0, 8, 11, 4, 9, 5, X, X, X, X, X, X, X},
    {0, 5, 4, 0, 1, 5, 2, 3, 11, X, X, X, X, X, X, X},
    {2, 1, 5, 2, 5, 8, 2, 8, 11, 4, 8, 5, X, X, X, X},
    {10, 3, 11, 10, 1, 3, 9, 5, 4, X, X, X, X, X, X, X},
    {4, 9, 5, 0, 8, 1, 8, 10, 1, 8, 11, 10, X, X, X, X},
    {5, 4, 0, 5, 0, 11, 5, 11, 10, 11, 0, 3, X, X, X, X},
    {5, 4, 8, 5, 8, 10, 10, 8, 11, X, X, X, X, X, X, X},
    {9, 7, 8, 5, 7, 9, X, X, X, X, X, X, X, X, X, X},
    {9, 3, 0, 9, 5, 3, 5, 7, 3, X, X, X, X, X, X, X},
    {0, 7, 8, 0, 1, 7, 1, 5, 7, X, X, X, X, X, X, X},
    {1, 5, 3, 3, 5, 7, X, X, X, X, X, X, X, X, X, X},
    {9, 7, 8, 9, 5, 7, 10, 1, 2, X, X, X, X, X, X, X},
    {10, 1, 2, 9, 5, 0, 5, 3, 0, 5, 7, 3, X, X, X, X},
    {8, 0, 2, 8, 2, 5, 8, 5, 7, 10, 5, 2, X, X, X, X},
    {2, 10, 5, 2, 5, 3, 3, 5, 7, X, X, X, X, X, X, X},
    {7, 9, 5, 7, 8, 9, 3, 11, 2, X, X, X, X, X, X, X},
    {9, 5, 7, 9, 7, 2, 9, 2, 0, 2, 7, 11, X, X, X, X},
    {2, 3, 11, 0, 1, 8, 1, 7, 8, 1, 5, 7, X, X, X, X},
    {11, 2, 1, 11, 1, 7, 7, 1, 5, X, X, X, X, X, X, X},
    {9, 5, 8, 8, 5, 7, 10, 1, 3, 10, 3, 11, X, X, X, X},
    {5, 7, 0, 5, 0, 9, 7, 11, 0, 1, 0, 10, 11, 10, 0, X},
    {11, 10, 0, 11, 0, 3, 10, 5, 0, 8, 0, 7, 5, 7, 0, X},
    {11, 10, 5, 7, 11, 5, X, X, X, X, X, X, X, X, X, X},
    {10, 6, 5, X, X, X, X, X, X, X, X, X, X, X, X, X},
    {0, 8, 3, 5, 10, 6, X, X, X, X, X, X, X, X, X, X},
    {9, 0, 1, 5, 10, 6, X, X, X, X, X, X, X, X, X, X},
    {1, 8, 3, 1, 9, 8, 5, 10, 6, X, X, X, X, X, X, X},
    {1, 6, 5, 2, 6, 1, X, X, X, X, X, X, X, X, X, X},
    {1, 6, 5, 1, 2, 6, 3, 0, 8, X, X, X, X, X, X, X},
    {9, 6, 5, 9, 0, 6, 0, 2, 6, X, X, X, X, X, X, X},
    {5, 9, 8, 5, 8, 2, 5, 2, 6, 3, 2, 8, X, X, X, X},
    {2, 3, 11, 10, 6, 5, X, X, X, X, X, X, X, X, X, X},
    {11, 0, 8, 11, 2, 0, 10, 6, 5, X, X, X, X, X, X, X},
    {0, 1, 9, 2, 3, 11, 5, 10, 6, X, X, X, X, X, X, X},
    {5, 10, 6, 1, 9, 2, 9, 11, 2, 9, 8, 11, X, X, X, X},
    {6, 3, 11, 6, 5, 3, 5, 1, 3, X, X, X, X, X, X, X},
    {0, 8, 11, 0, 11, 5, 0, 5, 1, 5, 11, 6, X, X, X, X},
    {3, 11, 6, 0, 3, 6, 0, 6, 5, 0, 5, 9, X, X, X, X},
    {6, 5, 9, 6, 9, 11, 11, 9, 8, X, X, X, X, X, X, X},
    {5, 10, 6, 4, 7, 8, X, X, X, X, X, X, X, X, X, X},
    {4, 3, 0, 4, 7, 3, 6, 5, 10, X, X, X, X, X, X, X},
    {1, 9, 0, 5, 10, 6, 8, 4, 7, X, X, X, X, X, X, X},
    {10, 6, 5, 1, 9, 7, 1, 7, 3, 7, 9, 4, X, X, X, X},
    {6, 1, 2, 6, 5, 1, 4, 7, 8, X, X, X, X, X, X, X},
    {1, 2, 5, 5, 2, 6, 3, 0, 4, 3, 4, 7, X, X, X, X},
    {8, 4, 7, 9, 0, 5, 0, 6, 5, 0, 2, 6, X, X, X, X},
    {7, 3, 9, 7, 9, 4, 3, 2, 9, 5, 9, 6, 2, 6, 9, X},
    {3, 11, 2, 7, 8, 4, 10, 6, 5, X, X, X, X, X, X, X},
    {5, 10, 6, 4, 7, 2, 4, 2, 0, 2, 7, 11, X, X, X, X},
    {0, 1, 9, 4, 7, 8, 2, 3, 11, 5, 10, 6, X, X, X, X},
    {9, 2, 1, 9, 11, 2, 9, 4, 11, 7, 11, 4, 5, 10, 6, X},
    {8, 4, 7, 3, 11, 5, 3, 5, 1, 5, 11, 6, X, X, X, X},
    {5, 1, 11, 5, 11, 6, 1, 0, 11, 7, 11, 4, 0, 4, 11, X},
    {0, 5, 9, 0, 6, 5, 0, 3, 6, 11, 6, 3, 8, 4, 7, X},
    {6, 5, 9, 6, 9, 11, 4, 7, 9, 7, 11, 9, X, X, X, X},
    {10, 4, 9, 6, 4, 10, X, X, X, X, X, X, X, X, X, X},
    {4, 10, 6, 4, 9, 10, 0, 8, 3, X, X, X, X, X, X, X},
    {10, 0, 1, 10, 6, 0, 6, 4, 0, X, X, X, X, X, X, X},
    {8, 3, 1, 8, 1, 6, 8, 6, 4, 6, 1, 10, X, X, X, X},
    {1, 4, 9, 1, 2, 4, 2, 6, 4, X, X, X, X, X, X, X},
    {3, 0, 8, 1, 2, 9, 2, 4, 9, 2, 6, 4, X, X, X, X},
    {0, 2, 4, 4, 2, 6, X, X, X, X, X, X, X, X, X, X},
    {8, 3, 2, 8, 2, 4, 4, 2, 6, X, X, X, X, X, X, X},
    {10, 4, 9, 10, 6, 4, 11, 2, 3, X, X, X, X, X, X, X},
    {0, 8, 2, 2, 8, 11, 4, 9, 10, 4, 10, 6, X, X, X, X},
    {3, 11, 2, 0, 1, 6, 0, 6, 4, 6, 1, 10, X, X, X, X},
    {6, 4, 1, 6, 1, 10, 4, 8, 1, 2, 1, 11, 8, 11, 1, X},
    {9, 6, 4, 9, 3, 6, 9, 1, 3, 11, 6, 3, X, X, X, X},
    {8, 11, 1, 8, 1, 0, 11, 6, 1, 9, 1, 4, 6, 4, 1, X},
    {3, 11, 6, 3, 6, 0, 0, 6, 4, X, X, X, X, X, X, X},
    {6, 4, 8, 11, 6, 8, X, X, X, X, X, X, X, X, X, X},
    {7, 10, 6, 7, 8, 10, 8, 9, 10, X, X, X, X, X, X, X},
    {0, 7, 3, 0, 10, 7, 0, 9, 10, 6, 7, 10, X, X, X, X},
    {10, 6, 7, 1, 10, 7, 1, 7, 8, 1, 8, 0, X, X, X, X},
    {10, 6, 7, 10, 7, 1, 1, 7, 3, X, X, X, X, X, X, X},
    {1, 2, 6, 1, 6, 8, 1, 8, 9, 8, 6, 7, X, X, X, X},
    {2, 6, 9, 2, 9, 1, 6, 7, 9, 0, 9, 3, 7, 3, 9, X},
    {7, 8, 0, 7, 0, 6, 6, 0, 2, X, X, X, X, X, X, X},
    {7, 3, 2, 6, 7, 2, X, X, X, X, X, X, X, X, X, X},
    {2, 3, 11, 10, 6, 8, 10, 8, 9, 8, 6, 7, X, X, X, X},
    {2, 0, 7, 2, 7, 11, 0, 9, 7, 6, 7, 10, 9, 10, 7, X},
    {1, 8, 0, 1, 7, 8, 1, 10, 7, 6, 7, 10, 2, 3, 11, X},
    {11, 2, 1, 11, 1, 7, 10, 6, 1, 6, 7, 1, X, X, X, X},
    {8, 9, 6, 8, 6, 7, 9, 1, 6, 11, 6, 3, 1, 3, 6, X},
    {0, 9, 1, 11, 6, 7, X, X, X, X, X, X, X, X, X, X},
    {7, 8, 0, 7, 0, 6, 3, 11, 0, 11, 6, 0, X, X, X, X},
    {7, 11, 6, X, X, X, X, X, X, X, X, X, X, X, X, X},
    {7, 6, 11, X, X, X, X, X, X, X, X, X, X, X, X, X},
    {3, 0, 8, 11, 7, 6, X, X, X, X, X, X, X, X, X, X},
    {0, 1, 9, 11, 7, 6, X, X, X, X, X, X, X, X, X, X},
    {8, 1, 9, 8, 3, 1, 11, 7, 6, X, X, X, X, X, X, X},
    {10, 1, 2, 6, 11, 7, X, X, X, X, X, X, X, X, X, X},
    {1, 2, 10, 3, 0, 8, 6, 11, 7, X, X, X, X, X, X, X},
    {2, 9, 0, 2, 10, 9, 6, 11, 7, X, X, X, X, X, X, X},
    {6, 11, 7, 2, 10, 3, 10, 8, 3, 10, 9, 8, X, X, X, X},
    {7, 2, 3, 6, 2, 7, X, X, X, X, X, X, X, X, X, X},
    {7, 0, 8, 7, 6, 0, 6, 2, 0, X, X, X, X, X, X, X},
    {2, 7, 6, 2, 3, 7, 0, 1, 9, X, X, X, X, X, X, X},
    {1, 6, 2, 1, 8, 6, 1, 9, 8, 8, 7, 6, X, X, X, X},
    {10, 7, 6, 10, 1, 7, 1, 3, 7, X, X, X, X, X, X, X},
    {10, 7, 6, 1, 7, 10, 1, 8, 7, 1, 0, 8, X, X, X, X},
    {0, 3, 7, 0, 7, 10, 0, 10, 9, 6, 10, 7, X, X, X, X},
    {7, 6, 10, 7, 10, 8, 8, 10, 9, X, X, X, X, X, X, X},
    {6, 8, 4, 11, 8, 6, X, X, X, X, X, X, X, X, X, X},
    {3, 6, 11, 3, 0, 6, 0, 4, 6, X, X, X, X, X, X, X},
    {8, 6, 11, 8, 4, 6, 9, 0, 1, X, X, X, X, X, X, X},
    {9, 4, 6, 9, 6, 3, 9, 3, 1, 11, 3, 6, X, X, X, X},
    {6, 8, 4, 6, 11, 8, 2, 10, 1, X, X, X, X, X, X, X},
    {1, 2, 10, 3, 0, 11, 0, 6, 11, 0, 4, 6, X, X, X, X},
    {4, 11, 8, 4, 6, 11, 0, 2, 9, 2, 10, 9, X, X, X, X},
    {10, 9, 3, 10, 3, 2, 9, 4, 3, 11, 3, 6, 4, 6, 3, X},
    {8, 2, 3, 8, 4, 2, 4, 6, 2, X, X, X, X, X, X, X},
    {0, 4, 2, 4, 6, 2, X, X, X, X, X, X, X, X, X, X},
    {1, 9, 0, 2, 3, 4, 2, 4, 6, 4, 3, 8, X, X, X, X},
    {1, 9, 4, 1, 4, 2, 2, 4, 6, X, X, X, X, X, X, X},
    {8, 1, 3, 8, 6, 1, 8, 4, 6, 6, 10, 1, X, X, X, X},
    {10, 1, 0, 10, 0, 6, 6, 0, 4, X, X, X, X, X, X, X},
    {4, 6, 3, 4, 3, 8, 6, 10, 3, 0, 3, 9, 10, 9, 3, X},
    {10, 9, 4, 6, 10, 4, X, X, X, X, X, X, X, X, X, X},
    {4, 9, 5, 7, 6, 11, X, X, X, X, X, X, X, X, X, X},
    {0, 8, 3, 4, 9, 5, 11, 7, 6, X, X, X, X, X, X, X},
    {5, 0, 1, 5, 4, 0, 7, 6, 11, X, X, X, X, X, X, X},
    {11, 7, 6, 8, 3, 4, 3, 5, 4, 3, 1, 5, X, X, X, X},
    {9, 5, 4, 10, 1, 2, 7, 6, 11, X, X, X, X, X, X, X},
    {6, 11, 7, 1, 2, 10, 0, 8, 3, 4, 9, 5, X, X, X, X},
    {7, 6, 11, 5, 4, 10, 4, 2, 10, 4, 0, 2, X, X, X, X},
    {3, 4, 8, 3, 5, 4, 3, 2, 5, 10, 5, 2, 11, 7, 6, X},
    {7, 2, 3, 7, 6, 2, 5, 4, 9, X, X, X, X, X, X, X},
    {9, 5, 4, 0, 8, 6, 0, 6, 2, 6, 8, 7, X, X, X, X},
    {3, 6, 2, 3, 7, 6, 1, 5, 0, 5, 4, 0, X, X, X, X},
    {6, 2, 8, 6, 8, 7, 2, 1, 8, 4, 8, 5, 1, 5, 8, X},
    {9, 5, 4, 10, 1, 6, 1, 7, 6, 1, 3, 7, X, X, X, X},
    {1, 6, 10, 1, 7, 6, 1, 0, 7, 8, 7, 0, 9, 5, 4, X},
    {4, 0, 10, 4, 10, 5, 0, 3, 10, 6, 10, 7, 3, 7, 10, X},
    {7, 6, 10, 7, 10, 8, 5, 4, 10, 4, 8, 10, X, X, X, X},
    {6, 9, 5, 6, 11, 9, 11, 8, 9, X, X, X, X, X, X, X},
    {3, 6, 11, 0, 6, 3, 0, 5, 6, 0, 9, 5, X, X, X, X},
    {0, 11, 8, 0, 5, 11, 0, 1, 5, 5, 6, 11, X, X, X, X},
    {6, 11, 3, 6, 3, 5, 5, 3, 1, X, X, X, X, X, X, X},
    {1, 2, 10, 9, 5, 11, 9, 11, 8, 11, 5, 6, X, X, X, X},
    {0, 11, 3, 0, 6, 11, 0, 9, 6, 5, 6, 9, 1, 2, 10, X},
    {11, 8, 5, 11, 5, 6, 8, 0, 5, 10, 5, 2, 0, 2, 5, X},
    {6, 11, 3, 6, 3, 5, 2, 10, 3, 10, 5, 3, X, X, X, X},
    {5, 8, 9, 5, 2, 8, 5, 6, 2, 3, 8, 2, X, X, X, X},
    {9, 5, 6, 9, 6, 0, 0, 6, 2, X, X, X, X, X, X, X},
    {1, 5, 8, 1, 8, 0, 5, 6, 8, 3, 8, 2, 6, 2, 8, X},
    {1, 5, 6, 2, 1, 6, X, X, X, X, X, X, X, X, X, X},
    {1, 3, 6, 1, 6, 10, 3, 8, 6, 5, 6, 9, 8, 9, 6, X},
    {10, 1, 0, 10, 0, 6, 9, 5, 0, 5, 6, 0, X, X, X, X},
    {0, 3, 8, 5, 6, 10, X, X, X, X, X, X, X, X, X, X},
    {10, 5, 6, X, X, X, X, X, X, X, X, X, X, X, X, X},
    {11, 5, 10, 7, 5, 11, X, X, X, X, X, X, X, X, X, X},
    {11, 5, 10, 11, 7, 5, 8, 3, 0, X, X, X, X, X, X, X},
    {5, 11, 7, 5, 10, 11, 1, 9, 0, X, X, X, X, X, X, X},
    {10, 7, 5, 10, 11, 7, 9, 8, 1, 8, 3, 1, X, X, X, X},
    {11, 1, 2, 11, 7, 1, 7, 5, 1, X, X, X, X, X, X, X},
    {0, 8, 3, 1, 2, 7, 1, 7, 5, 7, 2, 11, X, X, X, X},
    {9, 7, 5, 9, 2, 7, 9, 0, 2, 2, 11, 7, X, X, X, X},
    {7, 5, 2, 7, 2, 11, 5, 9, 2, 3, 2, 8, 9, 8, 2, X},
    {2, 5, 10, 2, 3, 5, 3, 7, 5, X, X, X, X, X, X, X},
    {8, 2, 0, 8, 5, 2, 8, 7, 5, 10, 2, 5, X, X, X, X},
    {9, 0, 1, 5, 10, 3, 5, 3, 7, 3, 10, 2, X, X, X, X},
    {9, 8, 2, 9, 2, 1, 8, 7, 2, 10, 2, 5, 7, 5, 2, X},
    {1, 3, 5, 3, 7, 5, X, X, X, X, X, X, X, X, X, X},
    {0, 8, 7, 0, 7, 1, 1, 7, 5, X, X, X, X, X, X, X},
    {9, 0, 3, 9, 3, 5, 5, 3, 7, X, X, X, X, X, X, X},
    {9, 8, 7, 5, 9, 7, X, X, X, X, X, X, X, X, X, X},
    {5, 8, 4, 5, 10, 8, 10, 11, 8, X, X, X, X, X, X, X},
    {5, 0, 4, 5, 11, 0, 5, 10, 11, 11, 3, 0, X, X, X, X},
    {0, 1, 9, 8, 4, 10, 8, 10, 11, 10, 4, 5, X, X, X, X},
    {10, 11, 4, 10, 4, 5, 11, 3, 4, 9, 4, 1, 3, 1, 4, X},
    {2, 5, 1, 2, 8, 5, 2, 11, 8, 4, 5, 8, X, X, X, X},
    {0, 4, 11, 0, 11, 3, 4, 5, 11, 2, 11, 1, 5, 1, 11, X},
    {0, 2, 5, 0, 5, 9, 2, 11, 5, 4, 5, 8, 11, 8, 5, X},
    {9, 4, 5, 2, 11, 3, X, X, X, X, X, X, X, X, X, X},
    {2, 5, 10, 3, 5, 2, 3, 4, 5, 3, 8, 4, X, X, X, X},
    {5, 10, 2, 5, 2, 4, 4, 2, 0, X, X, X, X, X, X, X},
    {3, 10, 2, 3, 5, 10, 3, 8, 5, 4, 5, 8, 0, 1, 9, X},
    {5, 10, 2, 5, 2, 4, 1, 9, 2, 9, 4, 2, X, X, X, X},
    {8, 4, 5, 8, 5, 3, 3, 5, 1, X, X, X, X, X, X, X},
    {0, 4, 5, 1, 0, 5, X, X, X, X, X, X, X, X, X, X},
    {8, 4, 5, 8, 5, 3, 9, 0, 5, 0, 3, 5, X, X, X, X},
    {9, 4, 5, X, X, X, X, X, X, X, X, X, X, X, X, X},
    {4, 11, 7, 4, 9, 11, 9, 10, 11, X, X, X, X, X, X, X},
    {0, 8, 3, 4, 9, 7, 9, 11, 7, 9, 10, 11, X, X, X, X},
    {1, 10, 11, 1, 11, 4, 1, 4, 0, 7, 4, 11, X, X, X, X},
    {3, 1, 4, 3, 4, 8, 1, 10, 4, 7, 4, 11, 10, 11, 4, X},
    {4, 11, 7, 9, 11, 4, 9, 2, 11, 9, 1, 2, X, X, X, X},
    {9, 7, 4, 9, 11, 7, 9, 1, 11, 2, 11, 1, 0, 8, 3, X},
    {11, 7, 4, 11, 4, 2, 2, 4, 0, X, X, X, X, X, X, X},
    {11, 7, 4, 11, 4, 2, 8, 3, 4, 3, 2, 4, X, X, X, X},
    {2, 9, 10, 2, 7, 9, 2, 3, 7, 7, 4, 9, X, X, X, X},
    {9, 10, 7, 9, 7, 4, 10, 2, 7, 8, 7, 0, 2, 0, 7, X},
    {3, 7, 10, 3, 10, 2, 7, 4, 10, 1, 10, 0, 4, 0, 10, X},
    {1, 10, 2, 8, 7, 4, X, X, X, X, X, X, X, X, X, X},
    {4, 9, 1, 4, 1, 7, 7, 1, 3, X, X, X, X, X, X, X},
    {4, 9, 1, 4, 1, 7, 0, 8, 1, 8, 7, 1, X, X, X, X},
    {4, 0, 3, 7, 4, 3, X, X, X, X, X, X, X, X, X, X},
    {4, 8, 7, X, X, X, X, X, X, X, X, X, X, X, X, X},
    {9, 10, 8, 10, 11, 8, X, X, X, X, X, X, X, X, X, X},
    {3, 0, 9, 3, 9, 11, 11, 9, 10, X, X, X, X, X, X, X},
    {0, 1, 10, 0, 10, 8, 8, 10, 11, X, X, X, X, X, X, X},
    {3, 1, 10, 11, 3, 10, X, X, X, X, X, X, X, X, X, X},
    {1, 2, 11, 1, 11, 9, 9, 11, 8, X, X, X, X, X, X, X},
    {3, 0, 9, 3, 9, 11, 1, 2, 9, 2, 11, 9, X, X, X, X},
    {0, 2, 11, 8, 0, 11, X, X, X, X, X, X, X, X, X, X},
    {3, 2, 11, X, X, X, X, X, X, X, X, X, X, X, X, X},
    {2, 3, 8, 2, 8, 10, 10, 8, 9, X, X, X, X, X, X, X},
    {9, 10, 2, 0, 9, 2, X, X, X, X, X, X, X, X, X, X},
    {2, 3, 8, 2, 8, 10, 0, 1, 8, 1, 10, 8, X, X, X, X},
    {1, 10, 2, X, X, X, X, X, X, X, X, X, X, X, X, X},
    {1, 3, 8, 9, 1, 8, X, X, X, X, X, X, X, X, X, X},
    {0, 9, 1, X, X, X, X, X, X, X, X, X, X, X, X, X},
    {0, 3, 8, X, X, X, X, X, X, X, X, X, X, X, X, X},
    {X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X}
};

边表用来获取在表面的顶点。而三角形表可以获得表面的三角片面序列。

Marcing Cube用以流体绘制

根据之前的SPH光滑核章节公式3.3:

我们计算得到的每个粒子的密度便可以作为标量场的值，该标量场也称为密度场。我们的粒子既然拥有自己的场，那么根据上面的方法便可以绘制流体的表面了。

MC类

class rxFace
{
public:
    vector<int> vert_idx;    // 顶点索引
    string material_name;    // 材质
    vector<glm::vec2> texcoords;    // 像素坐标
    int attribute;            //  属性
    
public:
    rxFace() : attribute(0) {}
    
    inline int& operator[](int i){ return vert_idx[i]; }
    inline int  operator[](int i) const { return vert_idx[i]; }
    
    // 按函数进行顶点访问
    inline int& at(int i){ return vert_idx.at(i); }
    inline int  at(int i) const { return vert_idx.at(i); }
    
    // 更改多边形顶点的数量
    void resize(int size)
    {
        vert_idx.resize(size);
    }
    
    // 返回多边形顶点的数量
    int size(void) const
    {
        return (int)vert_idx.size();
    }
    
    // 初始化
    void clear(void)
    {
        vert_idx.clear();
        material_name = "";
        texcoords.clear();
    }
};

struct RxVertexID{ //点的信息
    unsigned int newID;//索引
    double x,y,z;//位置
};

typedef std::map<unsigned int,RxVertexID> ID2VertexID;

struct RxTriangle{ //三角面的信息
    unsigned int pointID[3];//存有每个点的索引
};

typedef std::vector<RxTriangle> RxTriangleVector;

struct RxScalarField{//标量场
    unsigned int iNum[3];//x,y,z的大小
    glm::vec3 fWidth;//每个网格宽度
    glm::vec3 fMin;//最小点坐标
};

class rxMCMesh{
public:
    rxMCMesh();
    ~rxMCMesh();
    
    //从样本量生成三角形网格
    bool CreateMeshV(float *field, glm::vec3 min_p, double h, int n[3], float threshold,
                     vector<glm::vec3> &vrts, vector<glm::vec3> &nrms, vector<rxFace> &face);
    
    //从样本量生成等值面网格
    void GenerateSurfaceV(const RxScalarField sf, float *field, float threshold,
                         vector<glm::vec3> &vrts, vector<glm::vec3> &nrms, vector<int> &tris);
    
    //等值面创建成功则返回true
    bool IsSurfaceValid() const { return m_bValidSurface; }
    
    //删除表面
    void DeleteSurface();
    
    //用于网格划分的网格大小（在未创建网格的情况下，返回值为-1）
    int GetVolumeLengths(double& fVolLengthX, double& fVolLengthY, double& fVolLengthZ);
    
    //有关创建的网格的信息
    unsigned int GetNumVertices(void) const { return m_nVertices; }
    unsigned int GetNumTriangles(void) const { return m_nTriangles; }
    unsigned int GetNumNormals(void) const { return m_nNormals; }
    
private:
    unsigned int m_nVertices;    //点的个数
    unsigned int m_nNormals;    //法线个数
    unsigned int m_nTriangles;    //三角面个数
    
    ID2VertexID m_i2pt3idVertices;            //形成等值面的顶点列表(以边索引为key，等值面的点为value)
    RxTriangleVector m_trivecTriangles;       //形成三角形的顶点列表
    RxScalarField m_Grid;            //标量场信息
    
    const float * m_ptScalarField;     //保存标量值的样本量
    
    float m_tIsoLevel;   //阈值
    bool m_bValidSurface;            //表面是否生成成功
    
    // 边id
    unsigned int GetEdgeID(unsigned int nX, unsigned int nY, unsigned int nZ, unsigned int nEdgeNo);
    
    // 顶点ID
    unsigned int GetVertexID(unsigned int nX, unsigned int nY, unsigned int nZ);
    
    // 计算边缘上的等值面
    RxVertexID CalculateIntersection(unsigned int nX, unsigned int nY, unsigned int nZ,
                                     unsigned int nEdgeNo);
    
    // 通过网格边缘两端的隐式函数值的线性插值计算等值点
    RxVertexID Interpolate(double fX1, double fY1, double fZ1, double fX2, double fY2, double fZ2,
                           float tVal1, float tVal2);
    
    // 以输出形式存储顶点和网格几何信息
    void RenameVerticesAndTriangles(vector<glm::vec3> &vrts, unsigned int &nvrts, vector<int> &tris,unsigned int &ntris);
    
    // 顶点法线计算
    void CalculateNormals(const vector<glm::vec3> &vrts, unsigned int nvrts, const vector<int> &tris,unsigned int ntris,vector<glm::vec3> &nrms, unsigned int &nnrms);
    
};

我们首先定义一个rxFace结构体，来保存每个面内的信息。定义RxVertexID作为保存点的信息。定义ID2VertexID为以网格边做为索引，表面点信息作为值的map。定义了RxTriangle的三角片面结构体，用以保存该三角片面中三个顶点索引。RxTriangleVector则用vector保存了所有的三角片面。rxMCMesh类中便有以ID2VertexID 定义的类成员，与RxTriangleVector定义的类成员。

由于我们绘制三角形的时候，是先根据边表，判断点落在哪条边上，然后通过插值计算出具体位置，然后我们可以通过ID2VertexID类对象保存以当前边为索引，索引的值为当前点点具体位置。而由于我们的三角形表获取的信息是点在哪条边上，所以我们用RxTriangleVector类对象从三角形表获取的边信息，加上通过ID2VertexID类对象从边信息获取的点具体位置，便可以获得最终三角形片面的三个顶点的最终位置了。

我们主要观察函数实现：

rxMCMesh::rxMCMesh(){
    m_Grid.fMin=glm::vec3(0.0);
    m_Grid.fWidth = glm::vec3(0.0);
    m_Grid.iNum[0] = 0;
    m_Grid.iNum[1] = 0;
    m_Grid.iNum[2] = 0;

    m_nTriangles = 0;
    m_nNormals = 0;
    m_nVertices = 0;

    m_ptScalarField = nullptr;
    m_fpScalarFunc = 0;
    m_tIsoLevel = 0;
    m_bValidSurface = false;
}

rxMCMesh::~rxMCMesh(){
    DeleteSurface();
}

void rxMCMesh::DeleteSurface()
{
    m_Grid.fWidth[0] = 0;
    m_Grid.fWidth[1] = 0;
    m_Grid.fWidth[2] = 0;
    m_Grid.iNum[0] = 0;
    m_Grid.iNum[1] = 0;
    m_Grid.iNum[2] = 0;

    m_nTriangles = 0;
    m_nNormals = 0;
    m_nVertices = 0;

    m_ptScalarField = NULL;
    m_tIsoLevel = 0;
    m_bValidSurface = false;
}


//*从样本成三角形网格
//* @param [in]字段样本量
//* @param [in] min_p网格的最小坐标
//* @param [in] h网格的宽度
//* @param [in] n [3]网格编号（x，y，z）
//* @param [in]阈值阈值（隐含函数值网格化此值）
//* @param [in]方法网格生成方法（“mc”，“rmt”，“bloomenthal”）
//* @param [out] vrts顶点坐标
//* @param [out] nrms顶点正常
//* @param [out] tris mesh
//* @ retval true Mesh代成功
//* @retval false网格生成失败


bool rxMCMesh::CreateMeshV(float *field, glm::vec3 min_p, double h, int *n, float threshold, vector<glm::vec3> &vrts, vector<glm::vec3> &nrms, vector<rxFace> &face){
    if(field == nullptr) return false;

    RxScalarField sf;
    for(int i = 0; i < 3; ++i){
        sf.iNum[i] = n[i];
        sf.fWidth[i] = h;
        sf.fMin[i] = min_p[i];
    }

    vector<int> tris;
    GenerateSurfaceV(sf, field, threshold, vrts, nrms, tris);

    if(IsSurfaceValid()){
        int nv = (int)GetNumVertices();
        int nm = (int)GetNumTriangles();
        int nn = (int)GetNumNormals();


        for(int i = 0; i < nn; ++i){
            nrms[i] *= -1.0;
        }

        face.resize(nm);
        for(int i = 0; i < nm; ++i){
            face[i].vert_idx.resize(3);
            for(int j = 0; j < 3; ++j){
                face[i][j] = tris[3*i+(2-j)];
            }
        }

        return true;
    }

    return false;
}


void rxMCMesh::GenerateSurfaceV(const RxScalarField sf, float *field, float threshold,
                                vector<glm::vec3> &vrts, vector<glm::vec3> &nrms, vector<int> &tris){
    // 等值面生成
    if(m_bValidSurface){
        DeleteSurface();
    }

    m_tIsoLevel = threshold;
    m_Grid.iNum[0] = sf.iNum[0];
    m_Grid.iNum[1] = sf.iNum[1];
    m_Grid.iNum[2] = sf.iNum[2];
    m_Grid.fWidth = sf.fWidth;
    m_Grid.fMin = sf.fMin;
    m_ptScalarField = field;

    unsigned int slice0 = (m_Grid.iNum[0] + 1);
    unsigned int slice1 = slice0*(m_Grid.iNum[1] + 1);

    // 等値面の生成
    for(unsigned int z = 0; z < m_Grid.iNum[2]; ++z){
        for(unsigned int y = 0; y < m_Grid.iNum[1]; ++y){
            for(unsigned int x = 0; x < m_Grid.iNum[0]; ++x){
                // 计算网格中的顶点放置信息表参考索引
                unsigned int tableIndex = 0;
                if(m_ptScalarField[z*slice1 + y*slice0 + x] < m_tIsoLevel)
                    tableIndex |= 1;
                if(m_ptScalarField[z*slice1 + (y+1)*slice0 + x] < m_tIsoLevel)
                    tableIndex |= 2;
                if(m_ptScalarField[z*slice1 + (y+1)*slice0 + (x+1)] < m_tIsoLevel)
                    tableIndex |= 4;
                if(m_ptScalarField[z*slice1 + y*slice0 + (x+1)] < m_tIsoLevel)
                    tableIndex |= 8;
                if(m_ptScalarField[(z+1)*slice1 + y*slice0 + x] < m_tIsoLevel)
                    tableIndex |= 16;
                if(m_ptScalarField[(z+1)*slice1 + (y+1)*slice0 + x] < m_tIsoLevel)
                    tableIndex |= 32;
                if(m_ptScalarField[(z+1)*slice1 + (y+1)*slice0 + (x+1)] < m_tIsoLevel)
                    tableIndex |= 64;
                if(m_ptScalarField[(z+1)*slice1 + y*slice0 + (x+1)] < m_tIsoLevel)
                    tableIndex |= 128;

                
                if(edgeTable[tableIndex] != 0){
                    // 计算边上的顶点
                    if(edgeTable[tableIndex] & 8){
                        RxVertexID pt = CalculateIntersection(x, y, z, 3);
                        unsigned int id = GetEdgeID(x, y, z, 3);
                        m_i2pt3idVertices.insert(ID2VertexID::value_type(id, pt));
                    }
                    if(edgeTable[tableIndex] & 1){
                        RxVertexID pt = CalculateIntersection(x, y, z, 0);
                        unsigned int id = GetEdgeID(x, y, z, 0);
                        m_i2pt3idVertices.insert(ID2VertexID::value_type(id, pt));
                    }
                    if(edgeTable[tableIndex] & 256){
                        RxVertexID pt = CalculateIntersection(x, y, z, 8);
                        unsigned int id = GetEdgeID(x, y, z, 8);
                        m_i2pt3idVertices.insert(ID2VertexID::value_type(id, pt));
                    }

                    if(x == m_Grid.iNum[0] - 1){
                        if(edgeTable[tableIndex] & 4){
                            RxVertexID pt = CalculateIntersection(x, y, z, 2);
                            unsigned int id = GetEdgeID(x, y, z, 2);
                            m_i2pt3idVertices.insert(ID2VertexID::value_type(id, pt));
                        }
                        if(edgeTable[tableIndex] & 2048){
                            RxVertexID pt = CalculateIntersection(x, y, z, 11);
                            unsigned int id = GetEdgeID(x, y, z, 11);
                            m_i2pt3idVertices.insert(ID2VertexID::value_type(id, pt));
                        }
                    }
                    if(y == m_Grid.iNum[1] - 1){
                        if(edgeTable[tableIndex] & 2){
                            RxVertexID pt = CalculateIntersection(x, y, z, 1);
                            unsigned int id = GetEdgeID(x, y, z, 1);
                            m_i2pt3idVertices.insert(ID2VertexID::value_type(id, pt));
                        }
                        if(edgeTable[tableIndex] & 512){
                            RxVertexID pt = CalculateIntersection(x, y, z, 9);
                            unsigned int id = GetEdgeID(x, y, z, 9);
                            m_i2pt3idVertices.insert(ID2VertexID::value_type(id, pt));
                        }
                    }
                    if(z == m_Grid.iNum[2] - 1){
                        if(edgeTable[tableIndex] & 16){
                            RxVertexID pt = CalculateIntersection(x, y, z, 4);
                            unsigned int id = GetEdgeID(x, y, z, 4);
                            m_i2pt3idVertices.insert(ID2VertexID::value_type(id, pt));
                        }
                        if(edgeTable[tableIndex] & 128){
                            RxVertexID pt = CalculateIntersection(x, y, z, 7);
                            unsigned int id = GetEdgeID(x, y, z, 7);
                            m_i2pt3idVertices.insert(ID2VertexID::value_type(id, pt));
                        }
                    }
                    if((x==m_Grid.iNum[0] - 1) && (y==m_Grid.iNum[1] - 1))
                        if(edgeTable[tableIndex] & 1024){
                            RxVertexID pt = CalculateIntersection(x, y, z, 10);
                            unsigned int id = GetEdgeID(x, y, z, 10);
                            m_i2pt3idVertices.insert(ID2VertexID::value_type(id, pt));
                        }
                    if((x==m_Grid.iNum[0] - 1) && (z==m_Grid.iNum[2] - 1))
                        if(edgeTable[tableIndex] & 64){
                            RxVertexID pt = CalculateIntersection(x, y, z, 6);
                            unsigned int id = GetEdgeID(x, y, z, 6);
                            m_i2pt3idVertices.insert(ID2VertexID::value_type(id, pt));
                        }
                    if((y==m_Grid.iNum[1] - 1) && (z==m_Grid.iNum[2] - 1))
                        if(edgeTable[tableIndex] & 32){
                            RxVertexID pt = CalculateIntersection(x, y, z, 5);
                            unsigned int id = GetEdgeID(x, y, z, 5);
                            m_i2pt3idVertices.insert(ID2VertexID::value_type(id, pt));
                        }

                    // 多边形生成
                    for(unsigned int i = 0; triTable[tableIndex][i] != 255; i += 3){
                        RxTriangle triangle;
                        unsigned int pointID0, pointID1, pointID2;
                        pointID0 = GetEdgeID(x, y, z, triTable[tableIndex][i]);
                        pointID1 = GetEdgeID(x, y, z, triTable[tableIndex][i+1]);
                        pointID2 = GetEdgeID(x, y, z, triTable[tableIndex][i+2]);
                        triangle.pointID[0] = pointID0;
                        triangle.pointID[1] = pointID1;
                        triangle.pointID[2] = pointID2;
                        m_trivecTriangles.push_back(triangle);
                    }
                }
            }
        }
    }


    RenameVerticesAndTriangles(vrts, m_nVertices, tris, m_nTriangles);
    CalculateNormals(vrts, m_nVertices, tris, m_nTriangles, nrms, m_nNormals);

    m_bValidSurface = true;
}



/*
  *获得边缘ID
  * @param [in] nX，nY，nZ网格位置
  * @param [in] nEdgeNo边数
  * @return边缘ID
  */
unsigned int rxMCMesh::GetEdgeID(unsigned int nX, unsigned int nY, unsigned int nZ, unsigned int nEdgeNo)
{
    switch(nEdgeNo){
        case 0:
            return GetVertexID(nX, nY, nZ) + 1;
        case 1:
            return GetVertexID(nX, nY + 1, nZ);
        case 2:
            return GetVertexID(nX + 1, nY, nZ) + 1;
        case 3:
            return GetVertexID(nX, nY, nZ);
        case 4:
            return GetVertexID(nX, nY, nZ + 1) + 1;
        case 5:
            return GetVertexID(nX, nY + 1, nZ + 1);
        case 6:
            return GetVertexID(nX + 1, nY, nZ + 1) + 1;
        case 7:
            return GetVertexID(nX, nY, nZ + 1);
        case 8:
            return GetVertexID(nX, nY, nZ) + 2;
        case 9:
            return GetVertexID(nX, nY + 1, nZ) + 2;
        case 10:
            return GetVertexID(nX + 1, nY + 1, nZ) + 2;
        case 11:
            return GetVertexID(nX + 1, nY, nZ) + 2;
        default:
            // Invalid edge no.
            return -1;
    }
}


/*！
*获取顶点ID
* @param [in] nX，nY，nZ网格位置
* @return顶点ID
*/
unsigned int rxMCMesh::GetVertexID(unsigned int nX, unsigned int nY, unsigned int nZ)
{
    return 3*(nZ*(m_Grid.iNum[1] + 1)*(m_Grid.iNum[0] + 1) + nY*(m_Grid.iNum[0] + 1) + nX);
}


/*！
  * 通过插值计算边缘上的等值点（从样本量）
  * @param [in] nX，nY，nZ网格位置
  * @param [in] nEdgeNo边数
  * @return网格顶点信息
  */
RxVertexID rxMCMesh::CalculateIntersection(unsigned int nX, unsigned int nY, unsigned int nZ, unsigned int nEdgeNo)
{
    double x1, y1, z1, x2, y2, z2;
    unsigned int v1x = nX, v1y = nY, v1z = nZ;
    unsigned int v2x = nX, v2y = nY, v2z = nZ;

    switch(nEdgeNo){
        case 0:
            v2y += 1;
            break;
        case 1:
            v1y += 1;
            v2x += 1;
            v2y += 1;
            break;
        case 2:
            v1x += 1;
            v1y += 1;
            v2x += 1;
            break;
        case 3:
            v1x += 1;
            break;
        case 4:
            v1z += 1;
            v2y += 1;
            v2z += 1;
            break;
        case 5:
            v1y += 1;
            v1z += 1;
            v2x += 1;
            v2y += 1;
            v2z += 1;
            break;
        case 6:
            v1x += 1;
            v1y += 1;
            v1z += 1;
            v2x += 1;
            v2z += 1;
            break;
        case 7:
            v1x += 1;
            v1z += 1;
            v2z += 1;
            break;
        case 8:
            v2z += 1;
            break;
        case 9:
            v1y += 1;
            v2y += 1;
            v2z += 1;
            break;
        case 10:
            v1x += 1;
            v1y += 1;
            v2x += 1;
            v2y += 1;
            v2z += 1;
            break;
        case 11:
            v1x += 1;
            v2x += 1;
            v2z += 1;
            break;
    }

    // 获取边的两点坐标
    x1 = m_Grid.fMin[0]+v1x*m_Grid.fWidth[0];
    y1 = m_Grid.fMin[1]+v1y*m_Grid.fWidth[1];
    z1 = m_Grid.fMin[2]+v1z*m_Grid.fWidth[2];
    x2 = m_Grid.fMin[0]+v2x*m_Grid.fWidth[0];
    y2 = m_Grid.fMin[1]+v2y*m_Grid.fWidth[1];
    z2 = m_Grid.fMin[2]+v2z*m_Grid.fWidth[2];

    unsigned int slice0 = (m_Grid.iNum[0] + 1);
    unsigned int slice1 = slice0*(m_Grid.iNum[1] + 1);
    float val1 = m_ptScalarField[v1z*slice1 + v1y*slice0 + v1x];
    float val2 = m_ptScalarField[v2z*slice1 + v2y*slice0 + v2x];
    RxVertexID intersection = Interpolate(x1, y1, z1, x2, y2, z2, val1, val2);

    return intersection;
}

/*！
  *网格顶点，以输出格式存储的几何信息
  * @param [out] vrts顶点坐标
  * @param [out] nvrts顶点数
  * @param [out] tris三角形多边形几何信息
  * @param [out] ntris三角形多边形的数量
  */
void rxMCMesh::RenameVerticesAndTriangles(vector<glm::vec3> &vrts, unsigned int &nvrts, vector<int> &tris, unsigned int &ntris)
{
    unsigned int nextID = 0;
    ID2VertexID::iterator mapIterator = m_i2pt3idVertices.begin();
    RxTriangleVector::iterator vecIterator = m_trivecTriangles.begin();

    // 刷新点
    while(mapIterator != m_i2pt3idVertices.end()){
        (*mapIterator).second.newID = nextID;
        nextID++;
        mapIterator++;
    }

    // 刷新三角面.
    while(vecIterator != m_trivecTriangles.end()){
        for(unsigned int i = 0; i < 3; i++){
            unsigned int newID = m_i2pt3idVertices[(*vecIterator).pointID[i]].newID;
            (*vecIterator).pointID[i] = newID;
        }
        vecIterator++;
    }

    // 将所有顶点和三角形复制到两个数组中，以便可以有效地访问它们。
    // 复制点
    mapIterator = m_i2pt3idVertices.begin();
    nvrts = (int)m_i2pt3idVertices.size();
    vrts.resize(nvrts);
    for(unsigned int i = 0; i < nvrts; i++, mapIterator++){
        vrts[i][0] = (*mapIterator).second.x;
        vrts[i][1] = (*mapIterator).second.y;
        vrts[i][2] = (*mapIterator).second.z;
    }
    // 复制制作三角形的顶点索引。
    vecIterator = m_trivecTriangles.begin();
    ntris = (int)m_trivecTriangles.size();
    tris.resize(ntris*3);
    for(unsigned int i = 0; i < ntris; i++, vecIterator++){
        tris[3*i+0] = (*vecIterator).pointID[0];
        tris[3*i+1] = (*vecIterator).pointID[1];
        tris[3*i+2] = (*vecIterator).pointID[2];
    }

    //释放空间
    m_i2pt3idVertices.clear();
    m_trivecTriangles.clear();
}

/*！
  *顶点法线计算
  * @param [in] vrts顶点坐标
  * @param [in] nvrts顶点数
  * @param [in] tris三角形多边形几何信息
  * @param [in] ntris三角形多边形的数量
  * @param [out] nrms法线
  * @param [out] nnrms法线数（=顶点数）
  */
void rxMCMesh::CalculateNormals(const vector<glm::vec3> &vrts, unsigned int nvrts, const vector<int> &tris, unsigned int ntris,vector<glm::vec3> &nrms, unsigned int &nnrms)
{
    nnrms = nvrts;
    nrms.resize(nnrms);

    // 初始化法线.
    for( unsigned int i = 0; i < nnrms; i++){
        nrms[i][0] = 0;
        nrms[i][1] = 0;
        nrms[i][2] = 0;
    }

    // 计算法线.
    for( unsigned int i = 0; i < ntris; i++){
        glm::vec3 vec1, vec2, normal;
         unsigned int id0, id1, id2;
        id0 = tris[3*i+0];
        id1 = tris[3*i+1];
        id2 = tris[3*i+2];

        vec1 = vrts[id1]-vrts[id0];
        vec2 = vrts[id2]-vrts[id0];
        normal = cross(vec1, vec2);

        nrms[id0] += normal;
        nrms[id1] += normal;
        nrms[id2] += normal;
    }

    // 单位化法线.
    for( unsigned int i = 0; i < nnrms; i++){
        normalize(nrms[i]);
    }
}

其中GenerateSurfaceV函数则是我们的Marching Cube的应用过程。我们把密度场作为输入，遍历密度场的每一个网格点，判断其是否超过密度阈值（超过标1否则标0）。最后的tableIndex的八位2进制则保存了顶点索引。然后我们将tableIndex作为边表edgeTable的索引，用它获取其在网格的哪一条边上并用CalculateIntersection函数计算出线性插值，确定其位置，并得到该点的对应边编号。我们最后将该边编号作为索引，点位置作为值传入m_i2pt3idVertices。另外我们还用三角形表索引出每个网格内的三角形序列，并保存至m_trivecTriangles。有了每个三角片面的每个顶点，我们就能对他们求取法线了，函数CalculateNormals实现了该过程。

之后我们如果只要表面的顶点，只需要访问m_i2pt3idVertices内的所有点。若需要构建表面，只需要访问m_trivecTriangles即可。

到这里我们的MC类就全完成了，接下来的一小节我们只要更改一下我们之前的流体系统，为其添加mc的应用，那么我们的流体就不会再是一个个点，而是有自己的表面了。